# How should I report results of a likelihood ratio test?

I'm using likelihood ratio testing to assess whether a behavioral model is a better description of my data than a simpler (so called restricted) model.

How should results of such statistical tests be reported?

• You might have more luck getting an answer at CrossValidated. stats.stackexchange.com – Seanny123 Jun 26 '15 at 15:41
• Thanks, Seanny123, but I strongly doubt that CrossValidated will have a useful answer. My question has to do with reporting a common statistical test in psychology journals. – blz Jun 28 '15 at 14:30

General reporting recommendations such as that of APA Manual apply. One should report exact p-value and an effect size along with its confidence interval. In the case of likelihood ratio test one should report the test's p-value and how much more likely the data is under model A than under model B.

Example: The data is 7.3, 95% CI [6.8,8.1] times more likely under Model A than under Model B. The hypothesis that the data is equally likely under the two models was rejected with p=0.006.

The above statements already indicate that likelihood ratio test does not tell you which

model is a better description of my data

as the likelihood is $p(\mathrm{Data}|\mathrm{Model})$ and to learn which model is a better description of the data you need to compute $p(\mathrm{Model}|\mathrm{Data})$.

• Thanks! Isn't the distinction of $p(\mathrm{Data}|\mathrm{Model})$ vs $p(\mathrm{Model}|\mathrm{Data})$ somewhat academic? In principle, the probability of data given a model is closely related to the probability of a model given data, no? Colloquially speaking, is it really that incorrect to say that a likelihood ratio test helps us choose a better model for our data? And another question: I've run LRTs independently for 6 conditions over 20 subjects, so it's impractical to give precise p-values, CIs and probability ratios for each. How can I summarize these results? – blz Jun 29 '15 at 13:04
• @6 conditions: one should avoid multiple tests. It's cumbersome to present such results and you are running into problems with multiple comparisons. Instead formulate a single hypothesis about the pattern of results you expect to emerge across all 6 conditions and test it. If you don't have such a hypothesis then avoid testing and do exploratory analysis. For instance I would plot the log-likelihood for each condition. Differences on a logscale translate into multiplicative differences on the original scale. Then it's easy to derive "A is X times more likely than B" statements from such graph. – matus Jun 29 '15 at 13:47
• I'm not sure I understand in concrete terms. We expect model A to better fit data from any condition (2x3, i.e. congruency by SOA, as this is a Posner cueing task), so we run a LRT for each subject and condition. We find that each LRT result favors model A at p < .001. Could you please formulate your recommendations in concrete terms? What should I be doing differently, here? Thanks again -- your comments are very useful! – blz Jun 29 '15 at 13:55
• @blz, then just do a single likelihood ratio test on all data and all conditions. Strictly speaking, the influence SOA and congruency is not identifiable anyway. A voodoo solution would be to fit both models to each subject and each condition and then run multivariate regression with loglikelihood as DV and SOA and congruency as IV. – matus Jun 29 '15 at 15:50
• While it's strictly true that for a single likelihood it's the likelihood is of the data | model, the test keeps the data constant and the ratio solves this issue. It can be interpreted as a comparison of the models. – John Jun 6 '17 at 3:30

The likelihood ratio test is distributed as χ²with degrees of freedom = the change in degrees of freedom between the two models. So, to give an example dropping one parameter from a model, you would report it like this:

χ² (1) = 3.4, p = 0.065